Question

a transpose of a square matrix obtained by replacing the elements by their corresponding cofactors are
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Answer 1

An adjoint matrix is also called an adjugate matrix. In other words, we can say that matrix A is another matrix formed by replacing each element of the current matrix by its corresponding cofactor and then taking the transpose of the new matrix formed.

Answer 2

The transpose of a square matrix obtained by replacing the elements by their corresponding cofactors is called as the adjoint of the matrix A.

• Let A be a square matrix of order n×n, then the adjoint matrix of A is denoted by adj(A)
• An adjoint matrix is also called as an adjugate matrix

Example:

Let

        $A=\left[\begin{array}{ccc}3&-3&-1\\6&4&1\\0&5&3\end{array}\right]$

The cofactor of each element of the matrix A will be

       $Cof(a_{11})=(-1)^{(1+1)}\begin{vmatrix}4&1\\5&3\end{vmatrix} = 7$

       $Cof(a_{12})=(-1)^{(1+2)}\begin{vmatrix}6&1\\0&3\end{vmatrix} = -18$

       $Cof(a_{13})=(-1)^{(1+3)}\begin{vmatrix}6&4\\0&5\end{vmatrix} = 30$

       $Cof(a_{21})=(-1)^{(2+1)}\begin{vmatrix}-3&-1\\5&3\end{vmatrix} =4$

       $Cof(a_{22})=(-1)^{(2+2)}\begin{vmatrix}3&-1\\0&3\end{vmatrix} =9$

       $Cof(a_{23})=(-1)^{(2+3)}\begin{vmatrix}3&-3\\0&5\end{vmatrix} =-15$

       $Cof(a_{31})=(-1)^{(3+1)}\begin{vmatrix}-3&-1\\4&1\end{vmatrix} =1$

       $Cof(a_{32})=(-1)^{(3+2)}\begin{vmatrix}3&-1\\0&3\end{vmatrix} =-9$

       $Cof(a_{33})=(-1)^{(3+3)}\begin{vmatrix}3&-3\\6&4\end{vmatrix} =30$

The cofactor matrix of A is

       $Cofactor\ matrix\ of\ A=\left[\begin{array}{ccc}7&-18&30\\4&9&-15\\1&-9&30\end{array}\right]$

The adjoint of the matrix A is obtained by the transpose of the cofactor matrix of A i.e.,

       $Adj(A) = \left[\begin{array}{ccc}7&4&1\\-18&9&-9\\30&-15&30\end{array}\right]$